The Probability Thread

Sky_Poker

Hi

We thought we'd start this interesting thread to discuss probability / problem scenarios for things outside of poker with the hope that discussing these might help people understand probability in poker. It's also a fun subject to discuss (for some anyway!)

So for example have you heard of these?:

1) The Monty Hall Problem - see here
2) The Birthday Problem - see here

Do you think there are any other probability scenarious commonly misunderstood in the world of poker?

Thanks
Sky Poker 

Unknown

I spent four days convincing engineers on an industrial placement that the Monty Hall problem was real, to the point that they co-created a spreadsheet to prove me wrong. The spreadsheet proved the problem was correct after 2m+ iterations. I've seen much better ways to describe it than when I was taught it though.

The Birthday one is a classic that still boggles your mind even when you understand the maths behind it. The problem with probabilities of that one is that it can statistically take you a very long time to come across a group of 20 people where it is true!

Unknown

As a very rudimental demonstration I've used before when coaching, is approximating the % of a poker outcome to that of a dice roll - if you lose AA vs KK aipf, its the equivalent if rolling a dice with someone, and if they roll a 6 they win, and any other number you win. You don't lose your temper when they roll a 6, because you understand its a very real and potential outcome in that isolated event.

DAVEYZZ

Yep always swap....good example is deal or no deal when 2 boxes left and one has 250k in it..you have to swap!

Never seen the birthday one...I will try it at poker this weekend!

Unknown

Daveyzz, Deal or No Deal is not affected by the Monty Hall theory. The host has no input as to opening a box that holds a non-jackpot prize, and when there is no external influence the probabilities remain unchanged at 50%.

Slipwater

In Response to Re: The Probability Thread:

Daveyzz, Deal or No Deal is not affected by the Monty Hall theory. The host has no input as to opening a box that holds a non-jackpot prize, and when there is no external influence the probabilities remain unchanged at 50%.
Posted by CoxyLboro

Very true, Coxy, but psychologically most people would be inclined to swap. There is an inherent belief that a lot of contestants carry (based on nothing concrete whatsoever) that the money has to be in the other box.

Sky_Poker

In Response to Re: The Probability Thread:

As a very rudimental demonstration I've used before when coaching, is approximating the % of a poker outcome to that of a dice roll - if you lose AA vs KK aipf, its the equivalent if rolling a dice with someone, and if they roll a 6 they win, and any other number you win. You don't lose your temper when they roll a 6, because you understand its a very real and potential outcome in that isolated event.
Posted by CoxyLboro

Great example.

Unknown

Probability is a vile, near untameable beast. I still recalling arguing until blue in the face with my Year 6 teacher about the likelihood of an event - something along the lines of her insisting that it was a 0% probability that the sun would rise in the North Pole during winter. I think I argued that cosmic occurances could potenially change that, and that a catastrophic meteor impact could change the Earth's axial tilt sufficiently to do so, or that the Sun may displace due to a black hole etc etc. Just because these events are highly unlikely does not allow you to say its a 0% probability!

Pedantic perhaps, but if you teach kids the wrong thing from the start...

Tikay10

In Response to Re: The Probability Thread:

In Response to Re: The Probability Thread : Very true, Coxy, but psychologically most people would be inclined to swap. There is an inherent belief that a lot of contestants carry (based on nothing concrete whatsoever) that the money has to be in the other box.
Posted by Slipwater

Now I'm interested.

The Birthday one is incred, I still can't suss why it works, but it does.  

Sky_Poker

On the subject of percentages...

... isn't it odd when footballers (or whoever) say "we gave it 110% today".




Here is a glass - please fill it to 110%

aussie09

even though the probability of something is next to zero, given long enough, it will definitely happen.



 

F_Ivanovic

In Response to Re: The Probability Thread:

even though the probability of something is next to zero, given long enough, it will definitely happen.  
Posted by aussie09

Not true - it may still never happen, even with an infinite amount of time! Sure, the chance of it never happening might be infinitely small but it's still a possibility!

F_Ivanovic

I taught the missus the monty hall problem only this week - didn't quite work though because she wanted the goat instead of the car!!

MAXALLY

In Response to Re: The Probability Thread:

I taught the missus the monty hall problem only this week - didn't quite work though because she wanted the goat instead of the car!!
Posted by F_Ivanovic

...you are kidding right?

Swog

I'm in my 3rd year of a mathematics degree so when I get home later I'll see what I can think of  

aussie09

In Response to Re: The Probability Thread:

In Response to Re: The Probability Thread : Not true - it may still never happen, even with an infinite amount of time! Sure, the chance of it never happening might be infinitely small but it's still a possibility!
Posted by F_Ivanovic

it is true.  given an infinite number of events, it is very true.  in fact, definite.

chicknMelt

In Response to Re: The Probability Thread:

In Response to Re: The Probability Thread : it is true.  given an infinite number of events, it is very true.  in fact, definite.
Posted by aussie09

what about if there is a button I can press (and only me) that will blow up the world.

there is a small chance the world will get blown up

but it may never happen, and eventually I will die.

VespaPX

For those of you that have a lot of time on your hands.

http://en.wikipedia.org/wiki/Poker_probability

aussie09

In Response to Re: The Probability Thread:

In Response to Re: The Probability Thread : what about if there is a button I can press (and only me) that will blow up the world. there is a small chance the world will get blown up but it may never happen, and eventually I will die.
Posted by chicknMelt

hi CM,

unfortunately your life is finite.  if there were an unending line of CMs one of your descendents would press the button.

but please don't just yet.

Unknown

In Response to Re: The Probability Thread:

On the subject of percentages... ... isn't it odd when footballers (or whoever) say "we gave it 110% today". Here is a glass - please fill it to 110%
Posted by Sky_Poker

You can actually fill a container to slightly above its geometric volume with fluid.

http://physics.stackexchange.com/questions/45068/how-far-can-water-rise-above-the-edge-of-a-glass

I can't manage 110% of a glass that size, but maybe about 101.5%!
 
For a standard shot glass, you could be looking at ~10% increase though!

F_Ivanovic

In Response to Re: The Probability Thread:

In Response to Re: The Probability Thread : You can actually fill a container to slightly above its geometric volume with fluid. http://physics.stackexchange.com/questions/45068/how-far-can-water-rise-above-the-edge-of-a-glass I can't manage 110% of a glass that size, but maybe about 101.5%!   For a standard shot glass, you could be looking at ~10% increase though!
Posted by CoxyLboro

But the real question is... can you not only fill it to that % but then carry the glass to the table you are sat at without spilling any of it?

Unknown

I think most students attempt that feat on daily basis

Unknown

Had to google 0% probability events after this thread. Very interesting concept, that is dominated by scenarios that are either paradoxial (it will rain and not rain at the same time tomorrow at the same location) or events outside the possible range of results (chance I throw a 7 on a standard die).

Slipwater

In Response to Re: The Probability Thread:

In Response to Re: The Probability Thread : hi CM, unfortunately your life is finite.  if there were an unending line of CMs one of your descendents would press the button. but please don't just yet.
Posted by aussie09

...perish the thought.

GELDY

In Response to The Probability Thread:

2) The Birthday Problem  
Posted by Sky_Poker

Never heard it called a problem before!

Fairly easy maths, but as Coxy, I think, said, the answer is tough to believe if you don't want to.

But a good suggestion for this use.

But we should also invoke the spirit of Occam's razor

i.e.

of the competing hypotheses

i) the RNG is rigged

ii) the regs have a boom box

iii) i just don't understand the maths

we should apply the appropriate heuristic and end up choosing iii as the most appropriate response before we invoke the spirit of fraudulent sky poker programmers.

TeddyBloat

couple of poker paradoxes people may not grasp at first:

----

if

A />B

and

B />C

is

A />C?

in logic, yes.

in poker, no.

A = JTs

B = 22

C = AK

using pokerstove:

JTs />22 [54% v 46%]

22s />AK [54% v 46%]

JTs<AK  [40% V 60%]


------

bluff catching a river:

you are facing an all in bet, do you call knowing you will be beat 60% of the time. ie you are wrong more often than you are right?

starting stacks 150bb, you both have 50bb in the middle, on the rive villain overbet shoves all in for his and our remaing 100bb. you know enough about his ranges to know that your hand beats only 40% of his range. more often than not you will lose an extra 100bb and your ENTIRE stack.

action?

CALL!

you will lose our stack 60% of the time

40% of the time you will rake a pot of 300bb

your excpected stack size after calling then will be:

[0.4 x 300] + [0.6 x 0]

120 + 0

120bb

if you fold your stack size will always be 100

since

120 /> 100

we call despite knowing we are "wrong" more often then we are 'right'.

psychologically it is the hardest thing to do. we are losing our stack more often than not, and our edge only shows up after a LOT of volume. short-term we may get HAMMERED.

Swog

Lets consider the case when we go all in with AA and are up against 2-7, assume we run this 25 times and ignore rake.

Let's also assume that we win 5 in 6 times with this hand (83.3%), in our sample lets assume we win 21 times out of 25. We can assume for a significantly large sample size (i.e. >25) that this is normally distributed (google this).

We can work out a 97.5% confidence interval for the expected value (i.e. 97.5% chance that this interval contains the epectation).

We can work out the (sample) variance using a neat little formula and hence the standard variation (the sq. root of the variance). In our case we have that the sample variance is 7.66.

So using the fact that a 97.5% confidence interval is approximately 2 standard deviations away from the mean (which is +18.35 in our case) we get a 97.5% confidence interval for the expectation as [+3.4, +32]

We conclude that we can in fact have a negative expectation in our interval (i.e. values between +3.4 and +10)

Swog

Heres another more simplistic example. Assume we take 5 all ins with AA vs opponents 7-2 in a tournament and also assume that we are out of the tournament if we lose. Yes each individual flip will result in ~88% (5/6 ish) chance of surviving, but if we were to consider the chances of being knocked out of the competition in 5 of such All ins we see that the probability of winning all 5 all ins is,

(5/6)*(5/6)*(5/6)*(5/6)*(5/6) = 2/5 = 40% approximately.

So we conclude that we would in fact be knocked out 60% of the time if we were involved in 5 alll ins.

In fact, 

10 all ins => 16%

20 all ins => 2.6%

50 all ins => 0.01%

Swog

Let's now consider that a gambler puts £1 on red or black on a roulette wheel (ignore the 0), he'll double his stake if he loses and leave the table if he wins, he continues this forever. Now one might think this is a sure way of winning money? Absolutely not. 

If he loses 9 on the trot his next bet will be ~£1,000 (gettin expensive right?) lose 19 on the trot your next bet will be just over £1,000,000. 

So, unless you have an infinite (isn't possible) amount of money you'll come unstuck.

Swog

The second child paradox.

If you meet someone and they tell you “I have two children, and at least one of them is a boy”, what is the probability that the other child is a boy? (Hint: It’s not 1/2). It’s actually 1/3.

tomgoodun

In Response to Re: The Probability Thread:

The second child paradox. If you meet someone and they tell you  “I have two children, and at least one of them is a boy” , what is the probability that the other child is a boy? (Hint: It’s not 1/2). It’s actually  1/3 .
Posted by Swog

Is the other one a goat?